No definite viewpoint for the accelerating traveler?

[Alain2.7183]

If an accelerating traveler, at some given instant in his travels, is told that his question "How old is my home twin right now?" has no unique, definite answer, then wouldn't he have to regard that statement as implying that she must not really EXIST at that instant at all? If she DOES exist right then, wouldn't he have to believe that she must be DOING something definite right then?

 

[Nugatory]

If an accelerating traveler, at some given instant in his travels, is told that his question "How old is my home twin right now?" has no unique, definite answer, then wouldn't he have to regard that statement as implying that she must not really EXIST [b'at that instant[/B] at all? If she DOES exist right then, wouldn't he have to believe that she must be DOING something definite right then?

She is doing something definite at every instant, no questions or doubt there. We can even label each of those instants with the time that herr wristwatch reads at that instant, say things like "When her wristwatch read 3:00 she sneezed; when it read 3:01 she wiped her eyes, ..." and so forth.

But when the accelerated traveler speaks of what she's doing "right now", we have to ask him what "right now" means. Which time on her wristwatch is the accelerated traveler talking about when he says "right now"? That question has no unique definite answer; so "right now" has no unique definite meaning, and therefore the question "What is she doing right now?" also has no unique definite answer.

 

[ghwellsjr]

If an accelerating traveler, at some given instant in his travels, is told that his question "How old is my home twin right now?" has no unique, definite answer, then wouldn't he have to regard that statement as implying that she must not really EXIST at that instant at all? If she DOES exist right then, wouldn't he have to believe that she must be DOING something definite right then?

In addition to everything Nugatory said about the traveler's inability to assign a "right now" to the home twin, the home twin also cannot assign a "right now" to the accelerating traveler. And it's not because the traveler is accelerating. Even if he stops accelerating, the problem still exists. And even if he stops at a point distant from the home twin such that he is stationary with respect to the home twin the problem still exists.

But Special Relativity provides a way to deal with the problem by defining an Inertial Reference Frame (IRF). You can pick any IRF you want and then you can talk about "right now" in a meaningful way. But you can pick another IRF and have a different meaning to "right now" that is just as valid as the first one.

 

[Dale]

If an accelerating traveler, at some given instant in his travels, is told that his question "How old is my home twin right now?" has no unique, definite answer, then wouldn't he have to regard that statement as implying that she must not really EXIST at that instant at all? If she DOES exist right then, wouldn't he have to believe that she must be DOING something definite right then?

As Nugatory mentioned, the phrases "right now", "at that instant", "right then", etc. have no unique meaning. So no question with any such phrase has a unique answer. Whether the rest of the question is about their age, what they are doing, or their existence doesn't remove the ambiguity in the question.

Now, once you specify a coordinate system then the questions become uniquely defined. Frame variant quantities are perfectly legitimate things to ask questions about, you just have to be clear what frame they refer to.

 

[1977ub]

Once we're in GR, and we're comparing a "Far out in space" twin with a "surface of the earth" twin, do we still have these same ambiguities?

 

[PAllen]

Once we're in GR, and we're comparing a "Far out in space" twin with a "surface of the earth" twin, do we still have these same ambiguities?

Even more so. You can't compare vectors at a distance in GR; you can in SR.

 

[1977ub]

Even more so. You can't compare vectors at a distance in GR; you can in SR.

People speak with confidence about the rate of clocks in orbit vs on Earth.

But they can do so without having a definite opinion about which tick of a satellite is simultaneous with a particular time EST, I take it.

 

[PAllen]

People speak with confidence about the rate of clocks in orbit vs on Earth.

But they can do so without having a definite opinion about which tick of a satellite is simultaneous with a particular time EST, I take it.

Correct. You are using a family of clocks to establish a particular coordinate system to great precision. To make this coordinate system work, you find that precise adjustments are needed to the rate of orbiting clocks. The direct observables in this situation are sequences of signals sent from one 'clock' to another. This direct observable is invariant. It would be 'explained' differently in different coordinate systems, but the result of the observation would always be predicted to be the same.

 

[Nugatory]

People speak with confidence about the rate of clocks in orbit vs on Earth.

But they can do so without having a definite opinion about which tick of a satellite is simultaneous with a particular time EST, I take it.

Yes. One way of thinking about it: When EST changes over to EDT, we all reset our clocks but that doesn't change the rate at which the clocks tick.

If I want to compare the rate at which someone else's clock is ticking relative to my own, I don't need to worry about what time the other guy thinks it is, nor how much time he thinks has passed since some event in his past. All I need to do is count how many times my clock ticks in a given interval, count how many times his clock ticks in the same interval, and compare. The trick, and the place where the simultaneity convention comes into play, is in deciding what "the same interval" means (and I hope that alarm bells went off in your mind when you saw those words).

 

[1977ub]

All I need to do is count how many times my clock ticks in a given interval, count how many times his clock ticks in the same interval, and compare.

Yes the point observer rec'vs tick-pulses but these don't translate into an origination time without confidence on how far the pulses traveled.

Does the GR earth-surface observer experience the same ambiguity deciding the distance to an orbiting satellite that the AO SR observer does an RF source?

 

[PAllen]

Yes the point observer rec'vs tick-pulses but these don't translate into an origination time without confidence on how far the pulses traveled.

Does the GR earth-surface observer experience the same ambiguity deciding the distance to an orbiting satellite that the AO SR observer does an RF source?

Distance in SR or GR is defined by integrating invariant interval along spaceilike path, i.e. along a curve of a simultaneity convention. Thus, within these theories, distance has no possible meaning without a simultaneity convention.

 

[Alain2.7183]

Once we're in GR, and we're comparing a "Far out in space" twin with a "surface of the earth" twin, do we still have these same ambiguities?

In the case of the traditional "Twin Paradox", to make the GR explanation analogous to the SR explanation (via the equivalence principle), the inertial home twin needs to be considered to be floating in space, with no real gravitational fields anywhere.

In the several descriptions I've seen that use a fictitious gravitational field to resolve the twin paradox from the traveler's perspective, there was no ambiguity anywhere ... the procedure always gave a specific (unique) answer to the question of how much the home twin ages during the traveler's turnaround (according to the traveler). And that answer was always the same answer that is given by the SR analysis that uses the momentarily co-moving inertial reference frames. See, for example, the Wikipedia page on the twin paradox, and in particular, their section on the traveler's perspective.

 

[PAllen]

In the case of the traditional "Twin Paradox", to make the GR explanation analogous to the SR explanation (via the equivalence principle), the inertial home twin needs to be considered to be floating in space, with no real gravitational fields anywhere.

In the several descriptions I've seen that use a fictitious gravitational field to resolve the twin paradox from the traveler's perspective, there was no ambiguity anywhere ... the procedure always gave a specific (unique) answer to the question of how much the home twin ages during the traveler's turnaround (according to the traveler). And that answer was always the same answer that is given by the SR analysis that uses the momentarily co-moving inertial reference frames. See, for example, the Wikipedia page on the twin paradox, and in particular, their section on the traveler's perspective.

As has been pointed out to you in another thread, any specific method will give a specific answer. In no way does that mean there is a unique answer to the amount the distant twin ages during turnaround. The two methods you mention agree because pseudo-gravity is dependent on (metric expressed in) coordinates. The specific method you refer to uses coordinates based on the simultaneity of instantly comoving observers (even if this is not made explicit by the writer). Since they are both based on the same simultaneity convention, it is no surprise they agree. However, if a different simultaneity convention were used, you would get a different metric, and a different answer for distant twin age as function of traveling twin's clock; what must agree is the total differential aging. You should also be aware that for a somewhat more complex twin trajectory, you can't use either of these methods - because the lines of simultaneity intersect. No problem, just use a different convention.

The idea that 'one method is presented' implies there is a unique preferred answer, is your (invalid) inference. It is not stated in explanations using this simplest approach to pseudo-gravity.

 

[Alain2.7183]

You should also be aware that for a somewhat more complex twin trajectory, you can't use either of these methods - because the lines of simultaneity intersect.

I don't think that the intersections of the traveler's lines of simultaneity affect the usefulness or legitimacy of that coordinate system, FOR HIM. I think that the ONLY thing that is important to the traveler, is that at each instant in his life, his coordinate system tells him the current position and the current age of every object in the entire (assumed flat) universe. In particular, his coordinate system has no need to be a GR "chart" (as defined by Wald): GR charts DO need to be invertible, because they must be capable of knitting together the multiple charts that are necessary to cover the entire (curved) universe. There is no such requirement for the traveler in SR, because his single coordinate system covers the entire flat universe ... no "knitting" is required. And the fact that a spacetime point may not determine a unique age of the traveler is of no importance to the traveler at all: he would just say "That's strange, but it's just the way nature works, like it or not".

 

[PAllen]

I don't think that the intersections of the traveler's lines of simultaneity affect the usefulness or legitimacy of that coordinate system, FOR HIM. I think that the ONLY thing that is important to the traveler, is that at each instant in his life, his coordinate system tells him the current position and the current age of every object in the entire (assumed flat) universe. In particular, his coordinate system has no need to be a GR "chart" (as defined by Wald): GR charts DO need to be invertible, because they must be capable of knitting together the multiple charts that are necessary to cover the entire (curved) universe. There is no such requirement for the traveler in SR, because his single coordinate system covers the entire flat universe ... no "knitting" is required. And the fact that a spacetime point may not determine a unique age of the traveler is of no importance to the traveler at all: he would just say "That's strange, but it's just the way nature works, like it or not".

Any coordinates system has the requirement that it doesn't give two labels to the same point. That's got nothing to do with GR. As to physics, what meaning do you think there is to the statement:

Both at 3pm on my watch and at 4pm on my watch, an Earth home clock read 7pm ? (in between, it advanced to 8pm).

That is, can you describe any way at all to correlate this statement with any observation you could make? (You cannot; not only that, all direct observation contradicts such a description - the Earth clock is seen to move monotonically forward, throughout any traveler journey). Given that no observation correlates to this, why should a sane person believe it? It NOT a required implication of SR; in fact I recently researched that Einstein, for example, never used such lines of simultaneity in any of his SR or GR work. So you would posit that Einstein never understood SR?

 

[Nugatory]

I think that the ONLY thing that is important to the traveler, is that at each instant in his life, his coordinate system tells him the current position and the current age of every object in the entire (assumed flat) universe.

Which is something that a "coordinate system" in which the lines of simultaneity intersect fails to do, right?

 

[jbriggs444]

Which is something that a "coordinate system" in which the lines of simultaneity intersect fails to do, right?

I may have lost context here. As I understand it we are trying to imagine a coordinate system attached to the traveller which assigns a time coordinate to remote events by waiting until the traveller's hyper-plane of simultanity crosses the remote event and then using the traveller's "time now" as the time coordinate for the remote event.

So it seems to me that if lines of simultaneity intersect, that is not a "hole" in space time that is unmapped by the traveller's coordinate system. Rather it is an area of space time that is multiply mapped; assigned more than one time coordinate.

 

[PAllen]

I may have lost context here. As I understand it we are trying to imagine a coordinate system attached to the traveller which assigns a time coordinate to remote events by waiting until the traveller's hyper-plane of simultanity crosses the remote event and then using the traveller's "time now" as the time coordinate for the remote event.

So it seems to me that if lines of simultaneity intersect, that is not a "hole" in space time that is unmapped by the traveller's coordinate system. Rather it is an area of space time that is multiply mapped; assigned more than one time coordinate.

Multiple mapping is prohibited for coordinates by definition. Do you really think it makes sense to say that NYC exploded at 3pm on my watch and also at 4PM on my watch, even though I only see it explode once, and no observation I can make is consistent with it being simultaneous to two points on my world line? Instead, don't you think it is better to say that these completely unobservable lines of simultaneity that Einstein never used have limits on their applicability. They are useful (as are other simultaneity conventions) when they don't lead to absurdities.

(It would be different if there was some observation consistent with multiple simultaneity. But there is none.)

 

[jbriggs444]

Multiple mapping is prohibited for coordinates by definition. Do you really think it makes sense to say that NYC exploded at 3pm on my watch and also at 4PM on my watch, even though I only see it explode once, and no observation I can make is consistent with it being simultaneous to two points on my world line?

Yes, I do think it makes sense to say that NYC exploded at two distinct coordinates. And you are incorrect about this giving rise to inconsistencies. It's just a coordinate. It's not an observable.

 

[PAllen]

Yes, I do think it makes sense to say that NYC exploded at two distinct coordinates. And you are incorrect about this giving rise to inconsistencies. It's just a coordinate. It's not an observable.

There are accepted definitions of coordinates. They label a point on a manifold (physically, an event) once. Thus, such a system is not a coordinate system.

Tell me, why would I want to use such a non-coordinate system reflecting a non-observables in a way the does lead to nonsense that has no observable or logical basis? Instead, I can use any valid coordinate system to compute any observable, and conceptually model reality in a consistent way.

[edit: Why absurd? I know that NYC blows up once; every possible observation I can make indicates it blows up once; I have a plethora of valid coordinates consistent with SR that model it blowing up once. Why should I choose a method that constructs mathematically invalid coordinates and models it as blowing up at two different times of my history? Really??]

 

[jbriggs444]

There are accepted definitions of coordinates. They label a point on a manifold (physically, an event) once. Thus, such a system is not a coordinate system.

Not all definitions of coordinates require that the things being labelled be manifolds. Not all "coordinate systems" on a manifold need be bijective, topology-preserving or even cartesian.

That said, I can certainly understand how you would want those properties to hold for useful coordinate systems. And I can understand that you might want to adopt terminology requiring this to be the case.

Tell me, why would I want to use such a non-coordinate system reflecting a non-observables in a way the does lead to nonsense that has no observable or logical basis? Instead, I can use any valid coordinate system to compute any observable, and conceptually model reality in a consistent way.

I can use polar coordinates to refer to the north pole without worrying overmuch about nonsense ensuing. Mind you I agree that using such coordinates to label the north pole is one thing. Using them to model physics at the north pole would be more difficult.

Topology is not my strong suit, but what I think you are saying is that you want a "mathematically valid" coordinate system to be one that embodies a homeomorphism between a manifold and cartesian n-space.

Any coordinate system which assigns multiple coordinates to the same point cannot (of course) be a homeomorphism because it fails to be a bijection.

[edit: Why absurd? I know that NYC blows up once; every possible observation I can make indicates it blows up once; I have a plethora of valid coordinates consistent with SR that model it blowing up once. Why should I choose a method that constructs mathematically invalid coordinates and models it as blowing up at two different times of my history? Really??]

The question you posed did not ask whether you should use a coordinate system that happens to have multiple coordinates for a single event. You asked whether I thought that it would make sense. I think that it does make sense. It's not an example that lends itself easily to a coordinate system that labels the same event twice, but one can contrive a labelling that does so.

Suppose that I am driving east when the NYC blows up. Let's say that it blows up at 2:30 pm EST. I glance at my clock and see that it reads 1:30 pm CST. But I am not paying careful attention and don't know whether I've crossed the time zone line yet.

I can label the NYC blow up at both 1:30 or 2:30 using "my personal time zone" coordinates. This does not entail that NYC blew up twice.

 

[Nugatory]

I can label the NYC blow up at both 1:30 or 2:30 using "my personal time zone" coordinates.

Do you reset your watch as you're driving? If you do, you're changing from one coordinate system to another. If you don't, there's only one label you can assign to the blowup event, and that's the time on your watch when the blowup happens.

 

[jbriggs444]

Do you reset your watch as you're driving? If you do, you're changing from one coordinate system to another. If you don't, there's only one label you can assign to the blowup event, and that's the time on your watch when the blowup happens.

I have one hour of coordinate values to play with. If I choose to use them to double-label events, that's my business, not yours.

 

[PAllen]

The question you posed did not ask whether you should use a coordinate system that happens to have multiple coordinates for a single event. You asked whether I thought that it would make sense. I think that it does make sense. It's not an example that lends itself easily to a coordinate system that labels the same event twice, but one can contrive a labelling that does so.

Why do you think it makes sense to use a model that has a feature that is counter-factual to all observations, especially what there are a plethora of lmodels consistent with both observations and SR to choose from?

As for definitions, the following is one common definition of coordinates:

"In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space"

To a mathematician, the pole in polar coordinates is not covered by the coordinate system. In fact, this feature, in the case of a sphere, is the quintessential example used to show that there are manifolds such that no single coordinate system (patch) can cover the whole object.

 

[Nugatory]

I have one hour of coordinate values to play with. If I choose to use them to double-label events, that's my business, not yours.

As long as you don't call them "coordinates", sure. You can even call them "coordinates" if you want, but...

Q: If we call the tail a leg, how many legs does a horse have?
A: Four. Calling a tail a leg doesn't make it a leg.

 

[WannabeNewton]

Yes, I do think it makes sense to say that NYC exploded at two distinct coordinates. And you are incorrect about this giving rise to inconsistencies. It's just a coordinate. It's not an observable.

You are being quite rude with your later posts to PAllen for no reason at all as he is correcting a very absurd statement you are making. A coordinate map is an n - tuple of coordinate functions on an open subset of the manifold representing space - time. You are claiming the coordinate map can take a point in this region and map it to two different coordinates (the events that characterize the space - time aspect of the manifold for a given family of observers). This is not even a problem of physics, you are going against the very definition of a function which is a basic set theoretical object.

 

[Dale]

Not all definitions of coordinates require that the things being labelled be manifolds. Not all "coordinate systems" on a manifold need be bijective

Those used in GR do. If you disagree, please provide a good reference supporting the use of non-bijective coordinates in GR.

See ch 2. (especially around p. 34-37): http://arxiv.org/abs/gr-qc/9712019

 

[jbriggs444]

Why do you think it makes sense to use a model that has a feature that is counter-factual to all observations, especially what there are a plethora of lmodels consistent with both observations and SR to choose from?

Regarded as a mere labelling of events, it is not counter-factual.

As for definitions, the following is one common definition of coordinates:

"In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space"

As I read that definition, it requires that for every coordinate there is exactly one point. i.e. that the coordinate mapping be a function from coordinates to points, but not neccessarily either an injection (mapping to any given point by at most one coordinate) or a surjection (mapping to every point by at least one coordinate)

On the same page that you reference, one sees the following:

"Schemes for locating points in a given space by means of numerical quantities specified with respect to some frame of reference. These quantities are the coordinates of a point. To each set of coordinates there corresponds just one point in any coordinate system, but there are useful coordinate systems in which to a given point there may correspond more than one set of coordinates"

Emphasis mine.

 

[WannabeNewton]

Not all definitions of coordinates require that the things being labelled be manifolds. Not all "coordinate systems" on a manifold need be bijective, topology-preserving...

Where exactly did you learn about coordinate charts and manifolds? You have all the wrong ideas about topological manifolds.

 

[WannabeNewton]

Regarded as a mere labelling of events, it is not counter-factual.

It is utter nonsense.

As I read that definition, it requires that for every coordinate there is exactly one point. i.e. that the coordinate mapping be a function from coordinates to points, but not neccessarily either an injection (mapping to any given point by at most one coordinate) or a surjection (mapping to every point by at least one coordinate)

More nonsense. Let M be a topological manifold. \forall p\in M there exists a coordinate chart (U,\varphi ) where U\subseteq M is a neighborhood of p and \varphi :U\rightarrow \mathbb{R}^{n} is a homeomorphism.

 

[Dale]

On the same page that you reference, one sees the following:

"Schemes for locating points in a given space by means of numerical quantities specified with respect to some frame of reference. These quantities are the coordinates of a point. To each set of coordinates there corresponds just one point in any coordinate system, but there are useful coordinate systems in which to a given point there may correspond more than one set of coordinates"

Emphasis mine.

These are not valid coordinate charts in GR. Other disciplines may make use of such coordinates, but not GR.

 

[jbriggs444]

You are being quite rude with your later posts to PAllen for no reason at all as he is correcting a very absurd statement you are making. A coordinate map is an n - tuple of coordinate functions on an open subset of the manifold representing space - time. You are claiming the coordinate map can take a point in this region and map it to two different coordinates (the events that characterize the space - time aspect of the manifold for a given family of observers). This is not even a problem of physics, you are going against the very definition of a function which is a basic set theoretical object.

Rather than continue this distraction, I will bow out, acknkowledging that I am at the very least using standard terminology incorrectly for purposes of this context.

 

[WannabeNewton]

A function need not be injective mate but it cannot map a value in the domain to two values in the range which is what you are doing by assigning two different events to the same point on the manifold for a single observer.

 

[Alain2.7183]

I've been looking at a lot of the posts that Mentz recommended to me in another thread:

What is CADO?

See this topic

https://www.physicsforums.com/showthread.php?t=490163

I also did a forum search on "CADO", and found some more recent posts, including this one:

https://www.physicsforums.com/showpost.php?p=4041919&postcount=287

That post showed what someone who is going around and around in a circle, at constant speed, would say is the current age of some inertial person who is located some distance away from the circle. That post was interesting to me, because it is very similar to an example that I saw in a NOVA program called "the fabric of the cosmos". There, Brian Greene gave an example of someone on a planet in an extremely distant galaxy, who is riding a bicycle around and around in a small circle, and who says that for each of his loops, the time here on Earth is swinging back and forth over centuries! Brian Greene got that result by using the spatial three-dimensional "simultaneous time slices" of the sequence of inertial frames that are momentarily co-moving with the bicycle rider. Brian also has essentially the same example in his book that has the same title as the NOVA show.

As far as I can recall, in both the TV show and in his book, Brian didn't seem to be presenting his method as "just one of several different possible answers" to the question of "What is the current age of the inertial person, according to the accelerating person?". My impression was that he seemed to present it as THE answer.

On my CADO forum search, I also found this link,

https://sites.google.com/site/cadoequation/cado-reference-frame

that seems to give a pretty good summary of all the CADO stuff. The CADO equation explained in there gives the same result as Brian Greene's "momentarily co-moving inertial frame" method, but it's quicker and easier.

 

[PAllen]

As far as I can recall, in both the TV show and in his book, Brian didn't seem to be presenting his method as "just one of several different possible answers" to the question of "What is the current age of the inertial person, according to the accelerating person?". My impression was that he seemed to present it as THE answer.

This is Brian Greene in a pop-sci setting. Further, you are probably reading more into than Greene intended - I doubt he explicitly said this is the one valid way to look at things. A generic issue with pop-sci is that you select one simple, WOW point of view to describe about a more complex, multi-faceted situation. Also, if you want to argue by reference to authority, I could respond my noting that never in Einstein's life did he use such lines of simultaneity in either SR or GR.

 

[Dale]

Brian didn't seem to be presenting his method as "just one of several different possible answers" to the question of "What is the current age of the inertial person, according to the accelerating person?". My impression was that he seemed to present it as THE answer.

THE answer to what question?

If the question is "how can you do physics in a non-inertial frame" then it is not even an answer let alone THE answer. If the question is "what is the naive non-inertial simultaneity convention which most novices tend to adopt and which is useless for any actual physics" then I would agree that it is probably THE answer.